{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 优化与深度学习"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "本节将讨论优化与深度学习的关系，以及优化在深度学习中的挑战。在一个深度学习问题中，我们通常会预先定义一个损失函数。有了损失函数以后，我们就可以使用优化算法试图将其最小化。在优化中，这样的损失函数通常被称作优化问题的目标函数（objective function）。依据惯例，优化算法通常只考虑最小化目标函数。其实，任何最大化问题都可以很容易地转化为最小化问题，只需令目标函数的相反数为新的目标函数即可。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 优化与深度学习的关系\n",
    "虽然优化为深度学习提供了最小化损失函数的方法，但是本质上，优化与深度学习的目标是有区别的。在以前的章节（模型选择、欠拟合和过拟合）中，我们区分了训练误差和泛化误差。由于优化算法的目标函数通常是一个基于训练数据集的损失函数，优化的目标在于降低训练误差。而深度学习的目标在于降低泛化误差。为了降低泛化误差，除了使用优化算法降低训练误差外，还需要注意应对过拟合。\n",
    "\n",
    "本章中，我们只关注优化算法在最小化目标函数上的表现，而不关注模型的泛化误差。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 优化在深度学习中的挑战\n",
    "我们在3.1节（线性回归）中对优化问题的解析解和数值解做了区分。深度学习中绝大多数目标函数都很复杂。因此，很多优化问题并不存在解析解，而需要使用基于数值方法的优化算法找到近似解，即数值解。本书中讨论的优化算法都是这类基于数值方法的算法。为了求得最小化目标函数的数值解，我们将通过优化算法有限次迭代模型参数来尽可能降低损失函数的值。\n",
    "\n",
    "优化在深度学习中有很多挑战。下面描述了其中的两个挑战，即局部最小值和鞍点。为了更好地描述问题，我们先导入本节中实验需要的包或模块。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import sys\n",
    "from mpl_toolkits import mplot3d # 三维画图\n",
    "import numpy as np\n"
   ]
  },
  {
   "attachments": {
    "image.png": {
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    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 局部最小值\n",
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'f(x)')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
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      "text/plain": [
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     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "from IPython import display\n",
    "\n",
    "def use_svg_display():\n",
    "    display.set_matplotlib_formats('svg') # 用矢量图显示\n",
    "\n",
    "def set_figsize(figsize=(3.5, 2.5)):\n",
    "    use_svg_display()\n",
    "    plt.rcParams['figure.figsize'] = figsize # 设置图的尺寸\n",
    "\n",
    "def f(x):\n",
    "    return x * np.cos(np.pi * x)\n",
    "\n",
    "set_figsize((4.5, 2.5))\n",
    "x = np.arange(-1.0, 2.0, 0.1)\n",
    "fig, = plt.plot(x, f(x))\n",
    "fig.axes.annotate('local minimum', xy=(-0.3, -0.25), xytext=(-0.77, -1.0), arrowprops=dict(arrowstyle='->'))\n",
    "fig.axes.annotate('global minimum', xy=(1.1, -0.95), xytext=(0.6, 0.8), arrowprops=dict(arrowstyle='->'))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "深度学习模型的目标函数可能有若干局部最优值。当一个优化问题的数值解在局部最优解附近时，由于目标函数有关解的梯度接近或变成零，最终迭代求得的数值解可能只令目标函数局部最小化而非全局最小化。"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 鞍点\n",
    "刚刚我们提到，梯度接近或变成零可能是由于当前解在局部最优解附近造成的。事实上，另一种可能性是当前解在鞍点（saddle point）附近。举个例子，给定函数:\n",
    "![image.png](attachment:image.png)\n",
    "我们可以找出该函数的鞍点位置。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
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      "text/plain": [
       "<Figure size 324x180 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
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   ],
   "source": [
    "x = np.arange(-2.0, 2.0, 0.1)\n",
    "fig, = plt.plot(x, x**3)\n",
    "fig.axes.annotate('saddle point', xy=(0, -0.2), xytext=(-0.52, -5.0), arrowprops=dict(arrowstyle='->'))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x)')"
   ]
  },
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    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)\n",
    "我们可以找出该函数的鞍点位置。也许你已经发现了，该函数看起来像一个马鞍，而鞍点恰好是马鞍上可坐区域的中心。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'y')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\r\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\r\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\r\n",
       "<!-- Created with matplotlib (https://matplotlib.org/) -->\r\n",
       "<svg height=\"150.3pt\" version=\"1.1\" viewBox=\"0 0 265.5 150.3\" width=\"265.5pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\r\n",
       " <defs>\r\n",
       "  <style type=\"text/css\">\r\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\r\n",
       "  </style>\r\n",
       " </defs>\r\n",
       " <g id=\"figure_1\">\r\n",
       "  <g id=\"patch_1\">\r\n",
       "   <path d=\"M 0 150.3 \r\n",
       "L 265.5 150.3 \r\n",
       "L 265.5 0 \r\n",
       "L 0 0 \r\n",
       "z\r\n",
       "\" style=\"fill:none;\"/>\r\n",
       "  </g>\r\n",
       "  <g id=\"patch_2\">\r\n",
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      "text/plain": [
       "<Figure size 324x180 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = np.mgrid[-1: 1: 31j, -1: 1: 31j]\n",
    "z = x**2 - y**2\n",
    "\n",
    "ax = plt.figure().add_subplot(111, projection='3d')\n",
    "ax.plot_wireframe(x, y, z, **{'rstride': 2, 'cstride': 2})\n",
    "ax.plot([0], [0], [0], 'rx')\n",
    "ticks = [-1,  0, 1]\n",
    "plt.xticks(ticks)\n",
    "plt.yticks(ticks)\n",
    "ax.set_zticks(ticks)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在图的鞍点位置，目标函数在x轴方向上是局部最小值，但在y轴方向上是局部最大值。\n",
    "\n",
    "假设一个函数的输入为k维向量，输出为标量，那么它的海森矩阵（Hessian matrix）有k个特征值。该函数在梯度为0的位置上可能是局部最小值、局部最大值或者鞍点。\n",
    "\n",
    "1、当函数的海森矩阵在梯度为零的位置上的特征值全为正时，该函数得到局部最小值。\n",
    "\n",
    "2、当函数的海森矩阵在梯度为零的位置上的特征值全为负时，该函数得到局部最大值。\n",
    "\n",
    "3、当函数的海森矩阵在梯度为零的位置上的特征值有正有负时，该函数得到鞍点。\n",
    "\n",
    "随机矩阵理论告诉我们，对于一个大的高斯随机矩阵来说，任一特征值是正或者是负的概率都是0.5。那么，以上第一种情况的概率为 0.5k。由于深度学习模型参数通常都是高维的（k很大），目标函数的鞍点通常比局部最小值更常见。\n",
    "\n",
    "在深度学习中，虽然找到目标函数的全局最优解很难，但这并非必要。我们将在本章接下来的几节中逐一介绍深度学习中常用的优化算法，它们在很多实际问题中都能够训练出十分有效的深度学习模型。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 小结\n",
    "1、由于优化算法的目标函数通常是一个基于训练数据集的损失函数，优化的目标在于降低训练误差\n",
    "\n",
    "2、由于深度学习模型参数通常都是高维的，目标函数的鞍点通常比局部最小值更常见"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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